KURSCHAK 1981

Prove that $$AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R$ are any five points in a plane.

Let $n > 2$ be an even number. The squares of an $n\times n$ chessboard are coloured with $\frac12 n^2$ colours in such a way that every colour is used for colouring exactly two of the squares. Prove that one can place $n$ rooks on squares of $n$ different colours such that no two of the rooks can take each other.

For a positive integer $n$ , $r(n)$ denote the sum of the remainders when $n$ is divided by $1, 2,..., n$ respectively. Prove that $r(k) = r(k -1)$ for infinitely many positive integers $k$ .